About game theory

Game theory has become very fashionable. Its "tools", it is said, can be used to "solve problems" in social sciences - specially in economics and politics. Is this true ? The best way to answer this question is, first, to recall what a "game" is, in the game-theory sense of the word, and second, what is meant by the expression "solution" - or by "solution concept". After that it will be clear that game theory is, perhaps, great fun, but, surely, devoid interest for somebody who wants to understand the real world.

Games as models

All games, in game theory, are formed by three ingredients :

1. Individuals, called players, who want to maximize their payoffs (they are rational).

2. Sets, whose elements are called strategies. Each player has one set of strategies, and player i is supposed to choose one element s in his set of strategies, Si (i = 1, ..., n, if there are n players).

3. Rules, that associate an outcome of the game with each strategy profile (s1, ... , si, ... , sn), siÎSi, i = 1, ..., n, and that assign payoffs to each outcome. Rules are often represented by payoff functions. If gi(.) is the payoff function for player i, gi(s1, ... , si, ... , sn) gives his payoff when the profile of strategies is (s1, ... , si, ... , sn).

Which strategies will players then choose ? Before answering this question, it is necessary to precise what each player knows about others, and about the rules of the game (and thus, about payoffs in each outcome).Game theorists assume, at least for a start, that there is complete information : everybody knows everything about everybody, including that they are rational (all this being common knowledge) .It is very clear that there is no real life situation which fulfills these conditions (1, 2 and 3, plus complete information); it is why game theory models are only stories, without real data - as parlour games.

The fundamental point : unique and simultaneous choices

When conditions 1, 2 and 3 are fulfilled, players are supposed to choose SEPARATELY AND SIMULTANEOUSLY one element of their strategy set, and then THE GAME IS OVER : the outcome corresponding to these choices (and, thus, players’ payoffs) is determined. Assuming that choices are unique and made simultaneously is fundamental in game theory ; without this assumption, outcomes will not be well defined. But, again, it is quite difficult - or impossible - to find real life situations where such an assumption can be accepted. And, moreover, usual, or " popular ", presentations of game theory completely ignore it, specially when they explain that, thanks to game theory, it is possible to analyse " strategical behavior ", " conflict " or " cooperation ", with " threats " and " retaliations ", and so on. How can this be possible when players decide - separately and simultaneously - once, and only once ?

In fact, confusion proceeds here from two sources : first, game theorists have an interest to " sell " their models, by insinuating that they can explain many aspects of real life (like "strategic interactions", " conflicts ", etc.) ; the other source of confusion comes from the fact that it is possible to construct games fulfilling conditions 1, 2 and 3. with more than one move (sequential games and repeated games), but with simultaneous and unique choices. In this kind of game, strategies are " lists of instructions ", about actions chosen by players at each move (at each node of the game tree). Strategies are, thus, " conditional ", as they are of the kind : " if I am in this situation, then I will do that ; if I am in this other situation, then I will do that ; and so on ".

Each player then chooses one list of instructions (a strategy) in his strategy set (whose elements are lists of instructions), all players doing the same, at the same time. When all lists of instructions are chosen, and known by everybody, players - or some kind of referee - determine what happens at each move (" a path in the game tree ") and, thus, every player’s payoff.

For instance, consider the prisoners’ dilemma game, played twice. Each player’s strategy set has then four elements, like : " I won’t implicate my colleague at the first move ; at the second move, I will implicate him if he has implicated me at the first move but I won’t implicate him if he has not implicated me at the first move ", or : " I implicate my colleague at the first move ; at the second move, I implicate him again, whatever has he done at the first move ".

Intuition is here misleading : it seems natural to consider that, like in real life, players decide successively, at each move, after observing what has happened in the precedent moves (and then " learning " from that). But game models don’t proceed in this way, and we know why : players are supposed to choose simultaneously one, and only one, strategy (of the conditional type when there is more than one move). This is not intuitive at all and, again, there is no real life situation of this type; for instance, it is impossible to find an example of an oligopoly, an entry with deterrence, or any kind of competition with threats, cooperation, etc. reduced to unique and simultaneous choices - even if textbooks, or books alluding to game theory " analysis " and " results ", create confusion when they speak of " dynamics " about game models with many moves. Indeed, these books never present clearly what we have called the " fundamental point " (i.e. unique and simultaneous choices). If they did so, a normal person would stop reading, as there is no interest (except for fun) to continue with so counterintuitive, and misleading, stories.

Nash equilibrium as a "solution"

Nash equilibrium is game theorists’ main "concept of solution" ; in their books and papers they almost always discuss its’ existence, uniqueness, sensitivity to parameter variations, etc. Indeed, as is suggested by the word "equilibrium", people think that Nash equilibrium results from some kind of process - so that it seems to be "realistic". But this is completely false : Nash equilibrium, as all game theory solution concept supposes unique and simultaneous choices. It is different from other outcomes by the fact that each player anticipate correctly other’s choices. Nash equilibrium is not the outcome of some "objective" process, but results from (subjective) player’s beliefs. Where do these beliefs come from ? That is the problem.

Cournot’s model gives an example of how, often, game theorists and, always, textbooks justify in a wrong way Nash equilibrium, as a " solution " of a game. They sketch both reaction curves in the same figure, and then focus attention on the point where they intersect. Isn’t this point the " obvious " solution of the model ? Indeed, the only way to present the model in a correct manner is to draw one figure with one duopolist’s reaction curve (this is what this duopolist knows) and another figure with the other duopolist’s reaction curve. Each duopolist then chooses one point in his reaction curve, and there is no reason that these points are " precisely " the points where the two curves intersect (indeed, the probability that this happens is zero). It is then clear that Nash (or Cournot) equilibrium is not a prediction of the model, and that there is no reason to give it such importance. Often, textbooks present Cournot equilibrium as the result of a process : one duopolist makes an offer, at random ; the other " reacts " to this offer, and makes his own offer ; in his turn, the first duopolist " reacts " to this offer, and so on, until they reach the equilibrium (the point where reactions curve intersect). But this is nonsense, because rational duopolists will modify their reaction curves as the process goes on (as they notice that their competitor reacts to their offers), and thus the equilibrium (intersection point), changes during the process, as each of them observes how the other reacts (or plays) - equilibrium is " path dependent ", and thus indeterminate.

Indeed, when game theorists try to justify the importance they give to Nash equilibrium, they are forced to concede that rationality is not enough. Mas-Colell, White and Green (Microeconomic Theory) in their "Discussion of the concept of Nash Equilibrium", ask : "Why must it be reasonable to expect players’ conjectures to be correct ? Or, in sharper terms, why should we concern ourselves with the concept of Nash equilibrium ?" After explaining that "rationality need not lead players’ forecasts to be correct" (p 248), the only reason for this to happen is that Nash equilibrium is what Schelling called a "focal point", an outcome that has a "natural appeal" (it is evident for everybody) or results from a stable social convention" (p 249). Why not ? But then, is it not better, and more interesting, to study "social conventions" in the world around us, than to do a lot of complicated mathematics to "prove" existence of at least one Nash equilibrium in "toy models" ?

About " experiments "

Games, in game theory, are simple stories : they don’t describe (even approximately) observed situations. But it is always possible to make " experiments ", asking people to " play the game ", as in parlour games. There a lot of " experiments " of this kind. What are their conclusions ? That " real " people, in flesh and blood, often don’t react as the theory predicts (when it predicts something). Even in the case of the simple prisonner’s dilemma, not repeated, there is a minority of people who choose the strategy " don’t " implicate, and it seems that they are not rational. When the game is repeated only two or three times, few people choose what seems to be the " rational " strategy (" always implicate "). But the most famous example is the so-called " ultimatum game " : player A says to player B : "Somebody will give me a cake only if you accept to share it with me. Now, I propose to give you x% of the cake. Do you accept ? ".

If B is rational, he must accept even if x is tiny (it is better to have something than nothing). If A is rational, and thinks that B is rational too, he will then propose an x near zero : in this case, there is a quite clear prediction. But " experiments " don’t confirm it : in general, people like B don’t accept propositions if x is far from 50%, and people like A don’t propose a tiny x. Game theorists explain this by sense of equity : if people feel that the share is " too unfair ", they prefer to diminish their gains than to accept. Everybody agrees with them, but we can deduce that if this is true in a such simple situation, it must be true in a lot of real life situations, where people live together, interact - and where payoffs are not only monetary. Indeed, allmost all " experiments " in game theory don’t confirm the predictions of the theory, even when these predictions are well defined : people don’t act as they are supposed to do.Recent game theory litterature is full of another kind of " experiments " in the so-called, and fashionable, " evolutionary game theory ". The starting point of this theory, is that people don’t choose : each player is identified with a strategy (often, a conditional one). We are then at the opposite side of the starting point of game theory, which is to try to determine how rational people choose (or can choose) one of their (many) strategies.

It’s amazing to see that game theorists - so proud of their " rigor " - use the same words (game theory) to design completely different theories (even if they have some formal similarities). Indeed, in " evolutionary game theory " each individual is reduced to a strategy, and their are " tournaments " where strategies are confronted, two by two. Payoffs give the number of " offsprings ", who are at their turn confronted in " tournaments ", and so on, until some kind of " equilibrium ", where some (or all) strategies " survive ", is reached.Countless " experiments " of this type can be made : only a computer and some imagination are needed ! Strategies can be very sophisticated (including some kind of " learning ", decisions in each move depending on what happened before that move) or very simple. Theory doesn’t predict anything ; theorists only choose the strategies and rules of the " tournament " that they will play (thanks to computers), and then comment the " results " obtained - whatever they are. Mathematicians can try to find which strategies are " evolutionary stable " in this or that " tournament ", and so on. But it remains to prove that these new kind of " stories " are of any interest - in biology as in social science.

A game but surely not a " tool " for understanding real world

When somebody speaks of game theory and its " results ", or " insights ", first ask : " could you please give me a real life situation with FACTS AND DATA that can be described as a game, in game theory’s sense ? ". If the answer that you are given is of the kind : "theory always simplifies, it tries to explain ’stylised facts’, to understand what rational choices can be in different kinds of situations (catching some important aspects of real life)", then ask : "OK, but then, can you tell me what are the predictions of game theory, its proposed ’solutions’ ? ". If your interlocutor replies : "well, the first thing to do is to see if there is (at least) a Nash equilibrium ", then insist : "Do you mean that Nash equilibrium is the prediction of the model, the result of players’ rational choices, its ’solution’ ? ", and wait for the answer ... If it doesn’t come, or if it is confused, then close your eyes and your ears, and refuse all the figures and maths studying "properties" of Nash equilibria. And go study the world around us, just as it is.